A qubit can be formed in a two-level quantum system having two orthogonal basis states (or “working states”), denoted |0> and |1>, which are used to define a state ψ of the qubit as a superposition of the working states, namely α|0>+β|1>, where α and β are complex numbers satisfying |α|2+|β|2=1. A quantum computing algorithm consists of one or more successive operations that can be performed on a qubit in an initial state ψi to transform the qubit into a final state ωf.
A multi-level system providing a qubit typically contains additional states, for example, in the form of higher-energy excited states. During qubit operations, the state of the qubit (or multiple entangled qubits) may not necessarily always be confined to the phase space of the two working states of the qubit and so an admixture of the working states and the additional states can arise. This problem is usually referred to a “loss of fidelity” or “quantum leakage”. It may also be described as an intrinsic path for decoherence, i.e. related only to the qubit or the quantum system itself, not to its interaction with the environment.
Quantum leakage is an inherent property of almost all real-world quantum systems and is a fundamental problem. It is a significant issue for quantum computers because it can limit or prevent certain quantum algorithms from being executed.
Efforts have been made to improve (or “optimise”) quantum algorithms. However, these efforts have focused on using fewer operations and making operations shorter, as described, for example, in A Del Duce et al: “Design and optimisation of quantum logic circuits for a three-qubit Deutsch-Jozsa algorithm implemented with optically-controlled, solid-state quantum logic gates” http://arxiv.org/pdf/0910.1673.pdf and R. M. Fisher: “Optimal Control of Multi-Level Quantum Systems” http://mediatum.ub.tum.de/doc/1002028/1002028.pdf. These approaches, however, do not consider loss of fidelity and, from a mathematical point of view, are only concerned with phase space corresponding to the working states of the qubits.
A well-known technique for reducing the quantum errors, such as quantum leakage, is quantum error correction, as described in P. W. Shor: “Scheme for reducing decoherence in quantum computer memory”, Physical Review A, volume 52, page R2493 (1995). However, this technique requires additional components, e.g. additional qubits.
EP 2 264 653 A1 describes a symmetric gate arrangement to reduce the quantum leakage to excited states.
Clement H. Wong, M. A. Eriksson, S. N. Coppersmith and Mark Friesen: “High-fidelity singlet-triplet S-T qubits in inhomogeneous magnetic fields”, Physical Review B, volume 92, issue 4, article number 045403 (2015) proposes an optimized set of quantum gates for a singlet-triplet qubit in a double quantum dot with two electrons utilizing the S-T subspace. Qubit rotations are driven by an applied magnetic field and a field gradient provided by a micromagnet. Fidelity of the qubit as a function of the magnetic fields is optimized, taking advantage of “sweet spots” where rotation frequencies are independent of the energy level detuning, providing protection against charge noise. However, the probability of leakage into a non-working state, namely between 0.0005 to 0.012, is still quite high compared to benchmark values of 0.0001.